A combination of rapid deconstruction and movement of Factories, the construction of 'shanty-factories' in semi permanent structures, and the massive press-ganging or volunteering of Soviet Women from the local areas; there was also, of course, slave/prisoner labor.
Antony Beevor doesn't go into much depth about the process, but he does describe the program as militaristic in nature. Given the Soviet state's ability to ruthlessly collectivize and nationalize; I believe his choice of words are fitting. Soviet Industry was removed Eastwards as any persons not already on the front with technical training (often including students and Komonsol (sp?) members) were formed into Worker's Columns and put to work dismantling, or alternatively creating, new factories. Likewise, as mentioned before, the ability to militarize the operation is a key advantage; they simply mobilized the manpower to keep the factories running in their new locales. Often these "Factory Regiments" were made up almost completely of Soviet woman - something that the Germans had not yet considered for their own economy.
I'll leave you with a quote from Stalingrad that describes at least superficially the program of evacuation.
"[The Germans] also underestimated the ruthless programme of evacuation of industry to the Urals and the militarization of the workforce. Over 1,500 factories had been evacuated....and reassembled by armies of technicians slaving through the winter. Few factories had any heating. Many had no windows at first, or proper roofing. Once the production lines started, they never stopped, unless halted by breakdowns, power failures or shortages....Manpower posed less of a problem. The Soviet authorities simply drafted in new populations of workers..." (p. 223-24)
Apologies in advance for the rambling text below. It's too late in the evening for me to take much time over structuring this better. Let me talk a bit about the economic/planning aspect of the operation and then shout if there's anything that's not clear.
In the second half of 1941 over Soviet 1,500 industrial enterprises were stripped of their capital stock, fittings, resources and staff and relocated to the East. This amounted to approximately 38% of the industrial stock in the evacuation zone; some 13% of total pre-war Soviet industry was therefore saved. Along with the machines and resources, some ten million people were also evacuated in the face in the advancing Germans.
As to just how this happened, well, the entire process was by nature chaotic. Pre-war planning had allowed for the conversion of civilian factories for war production and there had been an effort to build up the industrial base outside of the vulnerable western regions. But the latter efforts hadn't achieved much (eg 63% of coal production and 58% of steel production lay in territories lost to the Germans) and absolutely no one expected to be fighting a war on the Volga. There were no real pre-war plans for this sort of catastrophe.
So the Soviets essentially had to react on the fly. The Committee for Evacuation was formed on 24 June (two days after the invasion) to coordinate the movement of sensitive industries east. There was no time to formulate an overall plan though, so the Committee laid down regulations for individual enterprises to follow. These priorities for evacuation were:
In terms of actually transporting this east, any and every means was used. Rail was obviously key - the evacuation alone used some 1.5 million rail cars in 1941 - but trucks were also used. The factories were stripped and loaded into whatever was available. The real miracle is that the Soviet rail network (which was also dealing with material flows in the opposite direction) coped with this burden remarkably well.
Actually getting the factories up and running again was obviously a challenge in itself. In this there was also some success: on reaching their destinations (ie the Urals, Siberia or Central Asia) a factory could be operational again in a matter of 4-6 weeks. This is obviously the best case but most enterprises were producing again by mid-1942. The economic dislocation remained severe.
Understandably, the actual state of the reconstructed factories was not ideal. Most were built from scratch and necessitated laying down of new infrastructure, allocation of new labour forces, etc. Accommodation and such comforts were not high on the priority list: cramped and unheated worker barracks would not have been the worst conditions.
Perhaps ironically, the deficiencies of Soviet 'planning' during the pre-war years meant that this sort of frenzied chaos was nothing new. Soviet industrialisation had always proceeded on the basis of mass mobilisations for 'shock' campaigns that threw resources at a problem; grossly inefficient in the pre-war years, it came into its own in 1941. The dire conditions greeting workers in the new factory sites in Siberia 1941 would not have been much different from those encountered during the FFYP - hurriedly constructed sheds and cold, austere dormitories.
Overall, it was a very impressive feat of coordination and mobilisation. To give an impression as to the speed with which it was accomplished, one first-hand account reports that, "In only 19 days there were removed from the Zaporozhstal' steelworks 16,000 wagons of vital machinery, including exceptionally valuable sheet-steel rolling-mill equipment. [...] The generator of the large turbine of the Zuevo power station was dismantled and loaded in eight hours. [...] The last train carrying the equipment arrived on 26 November and in two weeks time the first MiG plane was assembled."
Sources Alec Nove, An Economic History of the USSR; Eugene Zaleski, Stalinist Planning for Economic Growth, Davies and Harrison, The Economic Transformation of the Soviet Union